Collective Probabilistic Dynamical Modeling of Sleep Stage Transitions

Sergio A. Alvarez

and Carolina Ruiz

Department of Computer Science, Boston College, Chestnut Hill, MA 02467 U.S.A.

Department of Computer Science, Worcester Polytechnic Institute, Worcester, MA 01609 U.S.A.

Keywords:

Time Series, Clustering, Modeling, Markov, Data Mining, Sleep.

Abstract:

This paper presents a new algorithm for time series dynamical modeling using probabilistic state-transition

models, including Markov and semi-Markov chains and their variants with hidden states (HMM and HSMM).

This algorithm is evaluated over a mixture of Markov sources, and is applied to the study of human sleep stage

dynamics. The proposed technique iteratively groups data instances by dynamical similarity, while simultane-

ously inducing a state-transition model for each group. This simultaneous clustering and modeling approach

reduces model variance by selectively pooling the data available for model induction according to dynamical

characteristics. Our algorithm is thus well suited for applications such as sleep stage dynamics in which the

number of transition events within each individual data instance is very small. The use of semi-Markov models

within the proposed algorithm allows capturing non-exponential state durations that are observed in certain

sleep stages. Preliminary results obtained over a dataset of 875 human hypnograms are discussed.

1 INTRODUCTION

Sleep is divided into stages from all-night recordings

of physiological signals, particularly scalp EEG and

facial EOG (electro-oculography), following well-

established staging standards (Rechtschaffen and

Kales, 1968), (Iber et al., 2007). Stages span light

sleep (stages N1 / NREM1 and N2 / NREM2), deep

sleep (slow wave sleep, or SWS), and a stage tradi-

tionally associated with dreaming – Rapid Eye Move-

ment (REM) (dreams are known to occur during SWS

as well (Cavallero et al., 1992)). The temporal se-

quence of stage labels is known as a hypnogram.

See Fig. 1 for a sample hypnogram from the sleep

database used in the present paper.

0 100 200 300 400 500 600 700 800 900

SWS

NREM 2

NREM 1

REM

wake

Time (epochs)

Sleep stage

Figure 1: Sample hypnogram from the present study.

The dynamics of sleep stage transitions are af-

fected by overall health (Burns et al., 2008), (Bianchi

et al., 2010), making dynamics a clinically important

aspect of sleep structure. The study of sleep stage dy-

namics involves the construction of dynamical mod-

els of discrete time series. A challenge that arises

in this context is the scarcity of key events in the

data: each data instance (all-night hypnogram) con-

tains on the order of 10

individual sleep stage labels,

but only a small number of actual transitions between

stages. Because of this, the information in a full night

hypnogram may be insufﬁcient to adequately model

the dynamics of sleep stage transitions (Bianchi et al.,

2010). The present paper proposes a new approach

for addressing this problem, based on simultaneous

clustering and dynamical modeling of data. The ap-

plications of the proposed technique extend beyond

the study of sleep, to other domains that present in-

frequently changing discrete time series.

1.1 Related Work

Clustering for time-series data has been a topic of

great interest (e.g., (Liao, 2005)). Previous works

have addressed clustering of Markov chains (Ramoni

et al., 2001), (Cadez et al., 2003), hidden Markov

models (HMM) (Smyth, 1997), impulse-response

curves (Sivriver et al., 2011), or more general dynam-

ical models (Cadez et al., 2000). Some of these prior

works rely on modeling individual instances, for ex-

ample by constructing individual Markov chain mod-

els (Ramoni et al., 2001), or optimizing the parame-

209

A. Alvarez S. and Ruiz C..

Collective Probabilistic Dynamical Modeling of Sleep Stage Transitions.

DOI: 10.5220/0004243102090214

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2013), pages 209-214

ISBN: 978-989-8565-36-5

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

ters of an instance-speciﬁc ﬁt function (Sivriver et al.,

2011). Such an approach is not well-suited for the

event-sparse data in sleep studies, as the temporal in-

formation available per instance is insufﬁcient for re-

liable statistical modeling (Bianchi et al., 2010).

The simultaneous modeling and clustering strat-

egy of the present paper is similar to that of (Sivriver

et al., 2011) for gene expression. However, the clus-

tering step in (Sivriver et al., 2011) involves estima-

tion of instance-speciﬁc parameters, a process that is

subject to high variance in the presence of small data

instances as considered here. In other prior work,

the application domain provides abundant temporal

information for each instance, as in the web navi-

gation data of (Cadez et al., 2003). The more gen-

eral E-M framework on which (Cadez et al., 2003)

is based (Dempster et al., 1977) does allow for an ap-

proach that applies in the present context, as described

below in section 2. A related approach in which indi-

viduals are clustered, allowing multiple instances for

each individual, is pursued in (Cadez et al., 2000). A

relevant alternative view of model-based clustering in

terms of a bipartite graph that connects instances with

generative models as generalized cluster centroids,

using the generative data likelihood as a proximity

measure, is presented in (Zhong and Ghosh, 2003).

2 METHODS

2.1 Markov Mixture Data

Preliminary experiments were performedon data gen-

erated by a Markov chain mixture model. Two or

three Markov chains, M

,···M

(k = 2 or k = 3), were

used, each over a two-element state space that can be

loosely associated with wake and sleep states. The

initial state is assumed to be wake for all generated

sequences. For each integer i between 1 and a desired

total number of sequences, N, an equiprobable choice

was made among the Markov chains M

,···M

. The

selected model was then used to generate an observa-

tion sequence of the desired length, L, which was used

as the i-th output sequence of the mixture model. The

values N = 50 and L = 100 were used in most trials.

2.2 Human Sleep Data

875 anonymized human polysomnographic record-

ings were obtained with IRB approval from the Sleep

Clinic at Day Kimball Hospital in Putnam, Connecti-

cut, USA. The recordings were staged in 30-second

epochs by trained sleep technicians using the R & K

standard (Rechtschaffen and Kales, 1968). R & K

NREM stages 3 and 4 were then combined to obtain

a single slow wave sleep (SWS) stage. This proce-

dure yields stage labels that are known (Moser et al.,

2009) to closely approximate the more recently pro-

posed AASM staging standard (Iber et al., 2007).

2.2.1 Sleep Data Descriptions

Three different versions of the human sleep dataset

are considered in the present paper, each correspond-

ing to a different description of the hypnogram time-

series that comprise the dataset.

Uncompressed Dataset. The uncompressed data

description uses full-length sequences of the standard

stage labels wake, N1, N2, SWS, REM. The large di-

mensionality of the uncompressed description leads

to long running times for Algorithm 1, and makes

convergence more difﬁcult. For this reason, exper-

iments involving the uncompressed data description

required reduction of the size of the dataset through

random sampling. 105 instances were used.

WNR and WLD Datasets. In the two compressed

sleep data descriptions, each stage bout is replaced by

a single occurrence of the stage in question. For ex-

ample, the subsequence wake, wake, wake, N1, N1,

N2, SWS, SWS becomes N1, N2, SWS. The bout

duration information is stored separately. Additional

compression is then performed by reducing the num-

ber of distinct stages considered from ﬁve to three.

• The Wake/NREM/REM (WNR) dataset combines

the three stages N1, N2, SWS into a single NREM

stage, yielding the stages Wake, NREM, REM.

• The Wake/Light/Deep (WLD) dataset combines

the three stages N1, N2, REM into a single Light

sleep stage, yielding stages Wake, Light, SWS.

Use of the WNR and WLD datasets leads to a substan-

tial reduction in computing time as compared with

the uncompressed dataset, and facilitates convergence

of the CDMC Algorithm, allowing experiments to be

performed over the full set of 875 hypnograms.

2.3 The Collective Dynamical

Modeling-Clustering (CDMC)

Algorithm

The core of the approach proposed in the present

paper is the simultaneous clustering and dynamical

modeling technique described in pseudocode in Algo-

rithm 1. In the case of sleep, the instances of the input

dataset D will be sequences of sleep stage labels from

the datasets described in section 2.2.1.

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2.3.1 Main Steps in Algorithm 1

The proposed technique simultaneously learns a set

of dynamical models (cluster prototypes) and a cor-

responding cluster labeling, by alternating between

model estimation and clustering steps, terminating

when the cluster labelings change very little.

• Model estimation (

learnMLProtoypes

) learns a

maximum data likelihood dynamicalmodel M

for

each cluster C

• Clustering (

learnMLClusterLabels

): assigns

each instance x to the cluster c(x) having the

model M

c(x)

most likely to generate x.

2.3.2 Dynamical Model Types in Algorithm 1

Algorithm 1 encompasses not only standard HMM,

but also other types of dynamical models M for which

procedures are available for calculation of the gen-

erative likelihood P(x|M) and for maximum likeli-

hood model estimation. In particular, semi-Markov

models are included, which we are pursuing in work

in progress in order to capture the non-exponentially

distributed bout durations observed in certain human

sleep stages (Bianchi et al., 2010), (Lo et al., 2002).

2.4 Evaluation

Algorithm 1 was evaluated using hidden Markov

models (HMM) as the dynamical models, with the

Baum-Welch algorithm (e.g., (Rabiner, 1989)) for

HMM training in the

learnMLPrototypes

function,

the Rand index (Rand, 1971) to measure clustering

similarity in the stopping criterion, and a pseudo-

random initial cluster labeling c

. Fully observable

Markov chains were used as the dynamical models

in additional experiments over the compressed sleep

data representations (section 2.2.1). Results appear

in section 3. All implementations were carried out in

MATLAB



(The MathWorks, 2012).

2.4.1 Cluster Separation

Separation between clusters was measured by the log

likelihood margin (LLM) – the difference in log likeli-

hood between the ﬁrst and second highest likelihood

cluster labels for each instance. Higher mean LLM

values indicate better cluster separation.

2.4.2 Statistical Signiﬁcance

Population means were compared using a paired t-

test when the requisite normality assumption holds.

In other cases, a Wilcoxon rank sum test was used to

compare medians.

3 RESULTS

3.1 Markov Mixture Data

3.1.1 Two Generative HMM

Mixture data was obtained by an equiprobable selec-

tion between two generative HMM, each with two

states. For such HMM, the transition matrices are

completely determined by their values along the main

diagonal. Multiples of the identity matrix were used

for simplicity. 50 sequences of length 100 were gener-

ated per trial. 100 independent trials were performed.

Time to Convergence. The observed distribution

of the number of iterations for convergence of Algo-

rithm 1 with two clusters is nearly unimodal, with me-

dian and mode of 3 iterations, mean value of 3.72, and

standard deviation of 1.47. Over 90% of trials con-

verge in 5 or fewer iterations. With three clusters, me-

dian convergence time increases to 4 iterations, and

the 90th percentile increases to 7 iterations.

Variation with Initial Conditions. 100 trials were

performed with pseudorandom initial cluster la-

bels. HMM transition matrices T with diagonals

(T(1, 1),T(2,2)) of (0.6,0.6) and (0.75,0.75) were

used in the mixture model that generates the training

data. Mean ± std observed cluster centroids resulting

after convergence of Algorithm 1 are (0.66,0.64) ±

(0.036, 0.033) and (0.76,0.76) ± (0.020,0.021), re-

spectively, which ﬁt the generative model well.

Dependence on Separation between Generative

HMM. Fig. 2 shows clustering results obtained

when one of the two generative matrices has diago-

nal elements (0.6,0.6). The other matrix has diago-

nal elements (0.75, 0.75). Each instance is displayed

at the point (

T(1,1),

T(2,2)), where

T is the transi-

tion matrix for that instance as learned by the Baum-

Welch algorithm. The single instance for which the

CDMC algorithm (Algorithm 1) produces a labeling

error appears darker in the ﬁgure. The number of la-

beling errors increases as the separation between the

two generative HMM decreases: no labeling errors

occur when the generative diagonal values are 0.6,

0.85 instead of 0.6, 0.75, while many errors occur

with diagonal values 0.6, 0.65, for example.

3.1.2 Three Generative HMM. Determination of

the Number of Clusters.

Experiments were performed over mixture data pro-

duced by an equiprobable selection among 3 dist-

CollectiveProbabilisticDynamicalModelingofSleepStageTransitions

211

Algorithm 1: Collective Dynamical Modeling-Clustering (CDMC).

Input: An unlabeled time-series dataset D = {x = (a

(x)) | t = 1,2,3,...n}; a positive integer,

k, for the desired number of clusters; an initial guess c

: D → {1,···k} of the cluster label

(x) of each instance x ∈ D; parameter values, s, specifying the desired conﬁguration of the

models (e.g., number of states); and a real number

minSim

between 0 and 1 for the minimum

clustering similarity required for stopping.

Output: A set M

,···M

of generative dynamical models (with conﬁguration parameters s),

together with a cluster labeling c : D → {1· · · k} that associates to each data instance, x, the

index c(x) of a model M = M

c(x)

for which the generative likelihood

∏

x∈D

P(x|M

c(x)

) is as

high as possible.

CDMC(D,k,c

,s,

minSim

)

(1) c(x) = c

(x) for all x in D

(2) c

old

(x) = 0 for all x ∈ D

(3) while CLUSTERINGSIMILARITY(c,c

old

) <

minSim

(4) c

old

= c

(5) (M

,···M

) = LEARNMLPROTOYPES(D,k, c,s)

(6) c = LEARNMLCLUSTERLABELS(D,M

···M

)

(7) return M

,···M

Figure 2: CDMC results (0.6,0.75 self-transition probabili-

ties). Triangles indicate learned cluster models.

inct generative HMM. 50 sequences of length 100

were used in each of 50 independent trials. Using 2

clusters in the CDMC algorithm, the observed mean

LLM (2.4.1) is approximately 9.9, which corresponds

to a likelihood ratio of approximately 2 10

. With 3

clusters, mean LLM increases to 10.1, which is sig-

niﬁcantly greater than for 2 clusters as assessed by a

paired t-test using 50 paired trials (p < 0.02). Use

of a paired t-test is justiﬁed here because the LLM

distribution is close to normal except at the far tails,

as observed in a quantile-quantile plot (not shown

due to space restrictions). Specifying 4 clusters leads

to a statistically signiﬁcant decrease in mean LLM

(p < 0.001). See Table 1. Thus, the number of gen-

erative models can be determined by maximizing the

LLM in the clustering results. These facts show that

the CDMC algorithm is able to uncover the statistical

structure that underlies the data generation process.

Table 1: Log-likelihood margin. 3-HMM mixture data.

number of clusters 2 3 4

mean LLM 9.87 10.07 6.83

3.2 Human Sleep Data

We summarize here the results obtained using the

CDMC algorithm (Algorithm 1) on the human sleep

data described in section 2.2.

3.2.1 HMM over Uncompressed Sleep Sample

We applied Algorithm 1 to a sample of 105 instances

drawn randomly from the human sleep dataset de-

scribed in section 2.2, with k = 2 clusters and a pseu-

dorandom initial choice of cluster labels. The algo-

rithm converges in a dozen or so iterations of the main

loop on average. The models learned in one of these

runs are visualized in Fig. 3. HMM with 2 states were

used, with 5 possible emitted symbols corresponding

to sleep stages 1, 2, SWS, REM, and wakefulness.

The left subplot displays individual data instances as

the diagonal elements of the 2× 2 state transition ma-

trices learned from them by Baum-Welch, with mark-

ers indicating cluster membership. The middle and

right subplots display the emission probability matri-

ces for the HMM models of the two clusters; the two

rows of each emission matrix are represented by the

solid and dashed lines in the lower subplots.

Observations. As observed in the left subplot in

Fig. 3, the diagonal elements of the individual state

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0.9 0.95 1

0.85

0.9

0.95

trans(1,1)

trans(2,2)

1 2 SWS REM wake

0.5

cluster1

log P(stage | state)

1 2 SWS REM wake

0.5

cluster2

Figure 3: HMM transition matrices (left) and emission probabilities (right), Markov mixture data.

Table 2: Iterations to convergence. 10 trials, two clusters.

WNR 2 5 14 4 10 3 2 3 9 16

WLD 2 4 2 2 2 2 5 21 5 4

Table 3: HSMM transition matrices, two clusters (WNR).

0.0000 0.9565 0.0435 0.0000 1.0000 0.0000

0.7879 0.0000 0.2121 0.9979 0.0000 0.0021

0.7864 0.2136 0.0000 1.0000 0.0000 0.0000

transition matrices are very close to 1, making it dif-

ﬁcult to distinguish between clusters based on tran-

sition probabilities alone (diagonal mean and me-

dian differences are not statistically signiﬁcant by a

Wilcoxon rank sum test). This is due to the long

average duration of stage bouts in comparison with

the HMM clock period. In the remainder of the

present paper, we address this modeling disadvantage

of Markov dynamics by using compressed datasets

(section 2.2.1), in which repetitions are eliminated

from the stage sequences. In work in progress, this is-

sue is resolved as a by-product of using semi-Markov

models, which represent the durations of state vis-

its explicitly by their distributions, rather than by a

period-by-period coin ﬂip as Markov models do.

Comparing the subplots on the right in Fig. 3, we

see that the collective HMM for cluster 2 is more

likely to emit stage SWS than is the cluster 1 model.

The observed differences in stage SWS probabilities

are statistically signiﬁcant (p < 0.05, using a bino-

mial model). Thus, the emission probabilities provide

separation between the clusters. Within each cluster

model, the states have specialized to correspond to

particular combinations of sleep stages. For exam-

ple, only the states with solid lines in these plots have

nonzero wake emission probability (p < 0.05).

3.2.2 Observable Markov Chains over

Compressed Sleep Data Representations

Wake–NREM–REM Data Representation. In

contrast with Markov mixture data (section 3.1), for

WNR sleep data the observed LLM (section 2.4)

distribution deviates substantially from normality.

Table 4: HSMM transition matrices, two clusters (WLD).

0 1.0000 0.0000 0.0000 0.9997 0.0003

0.9734 0 0.0266 0.6919 0.0000 0.3081

0.5389 0.4611 0 0.4668 0.5332 0.0000

The median LLM for the WNR data is roughly 6.4,

which corresponds to a likelihood ratio of 600: the

maximum likelihood cluster is 600 times as probable

as the next best cluster. For a WNR sequence

of median length 34, this equates to 20% higher

generative probability per symbol (e

6.4/34

≈ 1.2).

Sample learned WNR transition matrices (Table 3)

show dynamical differences between clusters: higher

NREM to REM and REM to NREM probabilities in

cluster 1 (left matrix, middle and bottom rows).

Wake–Light–Deep Data Representation. Table 2

compares the WNR and WLD convergence times in

ten trials, with a minimum Rand index of 0.95 as the

stopping criterion. The median and mean of 3 and 4.9

iterations for WLD data are slightly lower than the

corresponding WNR values, 4.5 and 6.8.

Typical transition matrices obtained by Algo-

rithm 1 over WLD data appear in Table 4. A wake

state is followed by light sleep with near certainty

(top row). However, while the ﬁrst cluster exhibits

a very high probability of a light sleep to wake tran-

sition (left matrix, middle row), the second cluster

shows a substantial probability of transitioning from

light sleep to deep sleep (right matrix, middle row).

The distribution of the state immediately after a deep

sleep state (bottom row) is similar for the two clusters.

The LLM distribution for the WLD data is qualita-

tively similar to that for the WNR data. However, the

observed median LLM is approximately 3.5, corre-

sponding to a likelihood ratio of approximately 33, or

roughly 10% greater generative probability per sym-

bol for a typical WLD sequence of median length

37 (e

3.5/37

≈ 1.1). Thus, the WLD cluster separa-

tion is less pronounced than for the WNR data (c.f.

section 3.2.2). This suggests preferential use of the

WNR sleep data description in future work.

CollectiveProbabilisticDynamicalModelingofSleepStageTransitions

213

4 CONCLUSIONS; FUTURE

WORK

This paper has proposed a technique for dynamical

modeling of time-series with infrequent changes, and

has applied it to the study of human sleep data. The

technique, collective dynamical modeling and clus-

tering (CDMC), is based on adaptive pooling of data,

through iteration of clustering and dynamical mod-

eling steps. CDMC is a general algorithm that al-

lows a variety of probabilistic state space paradigms

(e.g., Markov chains, HMM, semi-Markov chains,

and HSMM) to be used as the dynamical models. Re-

sults over Markov mixture data show that the CDMC

algorithm converges rapidly, and that it successfully

identiﬁes the statistical structure underlying the data

generation process. Preliminary results have been ob-

tained over human sleep data, using a compressed

data representation that captures the temporal order-

ing of stage transitions but not the stage bout dura-

tions. These results demonstrate convergence of the

CDMC algorithm over real clinical data, with good

cluster separation. The clusters found are shown to be

characterized by distinct sleep-dynamical properties.

Work in progress by the authors builds on the

present paper by including detailed stage bout timing

information, using semi-Markov chains as the spe-

ciﬁc dynamical models in the CDMC algorithm. In

future work, there will be a need to systematically as-

sess convergence, as well as clustering stability with

respect to initial parameter values. The effect on

convergence of alternative strategies for initialization

should also be examined. The applications to sleep

dynamics of the CDMC algorithm proposed in the

present paper should be explored in greater detail.

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